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Theorem eunex 4219
Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.)
Assertion
Ref Expression
eunex  |-  ( E! x ph  ->  E. x  -.  ph )

Proof of Theorem eunex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dtru 4217 . . . . 5  |-  -.  A. x  x  =  y
2 alim 1548 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( A. x ph  ->  A. x  x  =  y ) )
31, 2mtoi 169 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  -.  A. x ph )
43exlimiv 1624 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  -.  A. x ph )
54adantl 452 . 2  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  ->  -.  A. x ph )
6 nfv 1609 . . 3  |-  F/ y
ph
76eu3 2182 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
8 exnal 1564 . 2  |-  ( E. x  -.  ph  <->  -.  A. x ph )
95, 7, 83imtr4i 257 1  |-  ( E! x ph  ->  E. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632   E!weu 2156
This theorem is referenced by:  reusv2lem2  4552  unnt  24919  amosym1  24937  alneu  28082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160
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